Mathematical Sciences: Applications of Nonstandard Analysis to Measure Theory, Potential Theory, and Related Topics

Loeb教授是应用数学中被称为非标准分析的基本概念的世界领先专家之一。这个概念最初是由逻辑学家提出的，是对无限大和无限小的经典概念的一个巧妙而严谨的发展。因此，非标准分析是莱布尼茨发明微积分时的哲学观点的数学证明。Loeb教授是在广泛的数学分析和应用中使用非标准方法来证明“标准”结果的主要实践者。勒布在非标准测度理论、非标准测度理论和非标准概率论方面的工作为概率论、数学物理和数学经济学做出了重要贡献。由首席研究员提出的勒布测度的概念，已经成为概率论、随机分析和势理论的标准工具。Loeb教授目前的建议有两个部分。首先，他提出将他的非标准积分理论与非标准凸包结合起来。展望了在巴拿赫空间和巴拿赫空间值随机过程中的应用。其次，他建议继续研究理想（勒布）边界的概念，这是为非常一般的势理论发展起来的。本研究对经典势理论具有重要的启示意义。

Professor Loeb is one of the world's leading experts in the applications of the fundamental foundational concept in mathematics called nonstandard analysis. This construct, first introduced by logicians, is an ingenious rigorous development of the classical concepts of infinitesimally large and small numbers. Nonstandard analysis, therefore, is a mathematical vindication of Leibnitz' philosophical viewpoint in his invention of the calculus. Professor Loeb is the leading practitioner in the use of nonstandard methods to prove "standard" results in a broad range of mathematical analysis and applications. Loeb's work in nonstandard measure theory and nonstandard measure theory and nonstandard probability theory has led to important contributions to probability theory, mathematical physics, and mathematical economics. The notion of Loeb measure, introduced by the principal investigator, has become a standard tool in probability theory, stochastic analysis, and potential theory. The current proposal by Professor Loeb has two parts. First, he proposes to combine his nonstandard integration theory with nonstandard convex hulls. Applications are envisaged to Banach spaces and Banach space valued stochastic processes. Secondly, he proposes to continue his work on the concept of ideal (Loeb) boundary, developed for very general potential theories. This research promises to have important implications for classical potential theory.